Statistical network models are used across the sciences and social sciences in settings such as modeling microbiomes and understanding protein interactions in biology and understanding friendships and strategic alliances in the social sciences. The $\beta$-stochastic blockmodel is a statistical network model that is useful in describing relational data that exhibit homophily, the tendency for certain individuals to group together. In the $\beta$-stochastic blockmodel, individuals are represented by nodes in an undirected graph which are grouped into blocks based on shared characteristics, and each edge is equipped with a parameter that measures the likelihood that the individuals represented by the nodes at each end interact. Ideally, one would like to determine the model parameters for the model that best fits a given dataset. We can measure the algebraic complexity of this problem by computing the maximum likelihood degree, the number of solutions to a set of likelihood equations associated with the model, for generic data. We explore the maximum likelihood degree for the $\beta$-stochastic blockmodel culminating in a multiplicative formula. Relevant background will be included in the talk, and the talk should be accessible to most.

This is joint work with Cash Bortner (California State University, Stanislaus), Elizabeth Gross (University of Hawaii at Manoa), Naomi Krawzik (Sam Houston State University), Christopher McClain (West Virginia University Institute of Technology), and Derek Young (Mount Holyoke College).

This work is a product of the 2023 Research Experiences for Undergraduate Faculty (REUF) program hosted at ICERM and the 2024 REUF continuation workshop at AIM, with support from NSF grants DMS-2015375, DMS-2015462, and DMS-1945584.

Link to arXiv pre-print: https://arxiv.org/abs/2410.06223